an old and basic question in string theory: What are the fundamental symmetry ... theory: First, in the past few years a covariant closed string field...

3 downloads 0 Views 154KB Size

Symmetries and Symmetry-Breaking in String Theory∗

Gregory Moore Department of Physics, Yale University, New Haven, CT 06511, USA

1. Introduction 1.1. Motivations This note summarizes a talk based on [7]. One motivation for this work is an old and basic question in string theory: What are the fundamental symmetry principles upon which theory should be based. If we understood the answer to this we would then go on to ask how the (on shell) symmetries are broken, and whether there is a notion of a “most symmetric background.” In asking these questions we have in mind analogous systems - Yang-Mills theories and general relativity. In the Yang-Mills example the answers to the above questions are known: the fundamental symmetry principles are local gauge invariance and Poincar´e invariance. Symmetries can be broken when a Higgs scalar φ takes a noninvariant expectation value. A symmetric background leaving the symmetries unbroken is hφi = 0. A second set of motivations comes from recent progress on two fronts in string theory: First, in the past few years a covariant closed string field theory (CSFT) has finally been constructed [13]. Unfortunately, the current formulation of the theory is rather complicated. One naturally hopes that a deeper understanding of the symmetry principles underlying CSFT will lead to simplification. A second piece of progress is one of the fruits of the matrix-model/2d gravity developments of 19891992. This was the realization that certain two-dimensional string backgrounds have large unbroken symmetries. Roughly speaking, they have a W∞ symmetry [12]. By a symmetry of the background we mean the following [12]. In CSFT a CFT C of c = 26 is a classical solution to the equations of motion. The string field Ψ represents deviations of the fields from this solution. CSFT is a gauge theory and the gauge transformation is Ψ → Ψ + QΛ + · · ·. Symmetries of a background C are therefore given by solutions to QΛ = 0. Dividing by transformations which act trivially we see that the symetries of a background C may be identified with a cohomology space. Classically, Ψ has ghost number 2 so the ghost number of Λ is one. Thus the symmetries of a background are identified with the ghostnumber 1 BRST G=1 (C). It can be shown that this cohomology space has a natural cohomology HBRST ∗

Summary of a talk given at SUSY93, Northeastern University, April 1, 1993.

structure of a Lie algebra. Indeed, that is a corollary of the full BV/Gerstenhaber algebra structure on H ∗ [2, 6, 10]. Examples: 1. Consider the standard bosonic string background R1,25 . The BRST cohomology is H 1 = R26 ⊕ R26 , corresponding to translations and “dual translations.” (The full Poincar´e invariance of this background arises from outer automorphisms of the CFT. It would be interesting to study examples of such outer automorphisms in other backgrounds.) 2. For certain string backgrounds in R2 one finds that H 1 is the Lie algebra of volume preserving diffeomorphisms of a 3-dimensional cone [12]. Comparing these two examples we see that Minkowksi space is extremely unsymmetric. These remarks lead one to ask if there are other extremely symmetric backgrounds in string theory. In this note we show that there are. 1.2. Method We will study symmetric backgrounds in the context of the standard 26dimensional bosonic string. The only new point is that we toroidally compactify all dimensions, including time. This might sound completely crazy, so let us offer three justifications for doing this. 1.) As is well-known, in toroidal compactifications one has the phenomenon of enhanced symmetry points (ESP’s) where there are length two right- or leftmoving vectors in the Narain lattice. At these points one has an enhanced gauge symmetry associated with a k = 1 WZW model. As we compactify more dimensions the enhanced gauge symmetries get bigger. Therefore, if we are searching for large symmetries we should take this to its logical conclusion and compactify all dimensions. 2.) One essential ingredient for the existence of the W∞ symmetry of 2d string backgrounds is the existence of a “timelike CFT” i.e., a CFT with negative dimension vertex operators. In the 2d string this role is played by the Liouville theory. Although the Liouville field is a Euclidean signature boson it is quite natural to consider real exponentials of this field, and one has conformal dimensions ∆(eαφ ) = √ 2 1 1 − 2 (α − 2) which are unbounded below. This example strongly suggests that if we are to understand the fundamental symmetry principles of string theory we will have to do peculiar things with the time coordinate. Compactification is the simplest solvable possibility. 3.) In the spacetime interpretation of toroidal compactification coordinates √ i i identified by X ∼ X + 2 2πRi have a breathing mode scalar in spacetime Φi (y) (y =coordinates of uncompactified spacetime) The zero-mode of this field corre¯ i . The radii may be identified with the scalar sponds to the vertex operator ∂X i ∂X VEV’s: Ri ∼ hΦi i. Thus hΦt i < ∞ might be unphysical but symmetric. We feel this is analogous to hφi = 0 in the Weinberg-Salam model: the background is unphysical,

but better suited to discovering the symmetries of the theory. 1.3. Summary of Results Let us give a brief, nontechnical summary of our results. In the remaining sections of the paper we will be more precise about details. First, there are some surprises in the generalization of the standard discussion of Narain compactification. The space of backgrounds B may be identified with the space of matrices E which have a decomposition into symmetric and antisymmetric parts E = G+ B such that G is a quadratic form of signature {−1, +125}. The first surprise is that B is not isomorphic to the homogeneous space O(1, 25) × O(25, 1)\O(26, 26), but is only an open proper subset. This means that the right action by the duality group (= O(26, 26; Z), with respect to an appropriate quadratic form) is not always defined. A much more important new feature is that the action of the duality group turns out to be ergodic. This means that the only group-invariant sets are of measure zero or of total measure. Moreover, the orbit of almost any background is dense. It follows that the Narain moduli space, which is, roughly speaking: O(1, 25) × O(25, 1) \O(26, 26)/O(26, 26; Z) is not a manifold in the ordinary sense, but is some kind of noncommutative manifold. Second, the phenomenon of enhanced gauge symmetries has some new features. In the Euclidean case there are isolated ESP’s with finite-dimensional enhanced symmetries. In the Minkowskian case the set of ESP’s is dense and the generic enhanced gauge symmetry is infinite-dimensional. Moreover, there is a distinguished point E∗ in the Narain moduli space of toroidal compactifications. This point has “maximal gauge symmetry” in the sense that all states in the BRST cohomology of the theory can be thought of as gauge bosons of the unbroken symmetry. The name “maximal symmetry” is slightly misleading. It does not mean that if E is an ESP then H 1 (E) has a Lie algebra embedding in H 1 (E∗ ). One may nevertheless wish to search for such a universal symmetry LU , that is, a Lie algebra which naturally includes H 1 (E) for all ESP’s E . This can be done and, curiously, involves a Fock space corresponding formally to a 52-dimensional “open string” with equal numbers of space and time directions. Third, one may wonder how we intend to apply any of these symmetries to spaces in which time is not compactified. The basic idea is that at a nonsymmetric background E ′ we may write E ′ = E∗ +∆E , and interpret ∆E as a symmetry-breaking term. This corresponds to spontaneous symmetry breaking in spacetime, since E ∼ hΦi. We may hope to use conformal perturbation theory to relate correlators and Ward identities at E∗ to correlators at E ′ . In particular, we propose a version of “broken symmetry Ward identities” below. We hope to apply these to explain the “high energy symmetries of string theory” discussed by D. Gross and E. Witten

[5, 11] in terms of the hyperbolic symmetries of compactified time. 2. Toroidal Compactifications 2.1. Algebraic Construction Toroidal compactifications in n + 1 dimensions are based on even self-dual lattices wrt a quadratic form D in 2n + 2 dimensions. We can construct such lattices from a generator matrix E whose columns define basis vectors for the lattice. Specifically we may take generator matrices: ˇ M = {E ∈ GL(2n + 2; IR) : E tr · D · E = D}

where D=

η 0

0 −η

ˇ ≡ D

0 1

1 0

(1)

(2)

ab and ηEab = Diag{1n+1} for Euclidean compactifications while ηM = Diag{−1, +1n} for Minkowskian compactifications. The set M is essentially an orthogonal group: ˇ = D and therefore M · S = O(D, IR) ∼ There is a matrix S such that SDS = Dˇ and S DS = ˇ R) ∼ S · M = O(D, = O(26, 26; IR). A central theorem of the subject states that all even unimodular lattices wrt D can be obtained from such E , so, dividing by the equivalence induced by integral basis change we see that the moduli space of lattices is

ˇ Z) ∼ ˇ IR)/O(D; ˇ Z) IL = M/O(D; = O(D;

(3)

For every point Γ ∈ IL we can define a CFT with statespace: HΓ = ⊕(pL ;pR )∈Γ FpL ⊗ F¯pR , where Fp is the Fockspace for the leftmovers with momentum p. For discussions of universal symmetry it is quite useful to use the notation preferred by mathemati− cians: HΓ = S(h− L ) ⊗ S(hR ) ⊗ C[Γ] where S denotes the symmetric algebra and the last factor is the group algebra of the lattice. Identifying CFT’s by left and right Lorentz transformations, which do not change conformal weights or correlators, we deduce that the Narain moduli space of toroidally compactified CFT’s is the double-coset: ˇ Z) N = O(η) × O(η) \M/O(D;

(4)

B = {E|E = G + B, signature(G) = η}

(5)

2.2. Sigma-model Construction In the sigma-model approach we begin with the space of toroidal backgrounds: and form the action: S=

1 2π

Z

0

2π

dσ

Z

¯ ν dτ ∂X µ Eµν ∂X

√ X µ ∼ X µ + 2 2π

(6)

which may be quantized in the standard way. Choosing left and right vielbeins for ¯ we find the mode expansions ¯ R = eR ∂X the metric G and defining ∂YL = eL ∂X, ∂Y

P ¯ a (z) = P β¯a z¯−n−1 define canonically normalized Heisenberg ali∂YLa (z) = βna z −n−1 , i∂Y n R b b gebras [βna , βm ] = η ab nδn+m,0 , [β¯na , β¯m ] = η ab nδn+m,0 . The lattice of zero modes is obtained

from the generator matrix

1 E(eL , eR , E) ≡ √ 2

eaµ L eaµ R

eaµ L Eµν tr −eaµ R Eµν

(7)

and an elementary calculation shows that E ∈ M. 2.3. New Features of Compactified Time The relation between the formulations can be summarized in the following diagram: ր

B

Mσ ψ

−→

֒→ ւ

IH

ց

M

N

ց

(8)

IL

ւ

The passage to moduli space through IL is the algebraic construction, the passage to moduli space through IH ≡ O(η) × O(η)\M is the sigma-model construction. Mσ is the space of matrices of the form in eq. (7). The map B → Mσ defined by E requires making a choice of vielbeins, so we should really speak of the well-defined map ψ. In the Euclidean case one can show that every matrix E ∈ M is of the form of eq. (7), thus identifiying B ∼ = IH. In the Minkowskian case this is not true. Strictly speaking, there are models which cannot be obtained from the sigma-model approach. Example: Consider compactification of 1 + 1 dimensions. Let G=

0 1

1 0

B=

0 1

−1 0

E=

0 2

0 0

(9)

Then E → E −1 does not make sense. Of course, the duality action on IH is wellˇ Z) on IH. The corresponding Mobius defined – it is just the right action of O(D; action on E does not make sense because duality takes us outside the “sigma-model set.” The fact that duality does not always act on B is not a very serious problem because there is always some duality transform which maps any point into the σmodel set Mσ . A much more serious observation is that the action of duality on IH and of left and right Lorentz transformations on IL is ergodic. Applying recentlydiscovered mathematical results concerning the action of noncompact groups on arithmetic quotients (like IL) one can deduce that for almost every point in IL the orbit under left or right Lorentz transformations is dense. Equivalently, for almost any background in IH the orbit of duality is dense. Example: Consider again 1 + 1-dimensional compactifications. Using the isomorphism of o(4) with sl(2) × sl(2) the moduli space of lattices can be written as

IL = SL(2, R)/SL(2, Z) × SL(2, R)/SL(2, Z) /(Z2 × Z2 )

(10)

The arithmetic quotient SL(2, R)/SL(2, Z) is the unit tangent bundle (in the Poincar´e metric) over the modular curve Σ = SO(2)\SL(2, R)/SL(2, Z). In Euclidean compactifications we must divide IL by SO(2) × SO(2). The left-action of SO(2) on SL(2, R)/SL(2, Z) is simply rotation in the fibers. In the Minkowskian case we must divide IL by SO(1, 1) × SO(1, 1) . The left-action of SO(1, 1) on SL(2, R)/SL(2, Z) is left-multiplication by cosh t sinh t

sinh t cosh t

(11)

This action is simply geodesic flow in the unit tangent bundle. The typical geodesic on Σ is dense. Thus we see that the moduli space of conformal field theories is a nonstandard space. It is a typical example of a noncommutative manifold. 3. Enhanced Gauge Symmetries 3.1. Enhanced Symmetry Points Recall that the unbroken symmetry at a background is the ghost number one cohomology. Since HΓ is a sum of Fock spaces this is easily calculated: 1

H (HΓ ) = ⊕pL ,pR

# 0 ¯ 1 ¯ H (FpL ) ⊗ H (FpR ) ⊕ H (FpL ) ⊗ H (FpR )

"

0

1

(12)

Since H ∗ (Fp ) = 0 for p2 ∈/ {2, 0, −2, −4, . . .} we see that for generic points in IL H 1 = R26 ⊕ R26 : the generic background is just as symmetric as Minkowski space. Let us define a background Γ ∈ IL to be an enhanced symmetry point (ESP) if ∃ (pL ; 0) ∈ Γ or (0; pR ) ∈ Γ with p2L or p2R ∈ {2, 0, −2, −4, . . .} There are infinite-dimensional unbroken gauge symmetries at a dense √set of points in IL. This is elementary: Consider lattices obtained from E such that 2E is a rational matrix. Moreover, for the generic such point the purely left- and rightmoving sublattices (γL ; 0) ⊕ (0; γR) ⊂ Γ will have maximal rank. In particular they will have an infinite number of points inside the forward and backward light cone and therefore the unroken symmetry algebras are typically infinite-dimensional. The unbroken symmetries belong to a class of Lie algebras known as “generalized KacMoody Lie algebras” [1]. The importance of these algebras in string theory has been emphasized by Goddard and Olive [4]. We will refer to these infinite-dimensional gauge symmetries as “hyperbolic symmetries.” † 3.2. A Distinguished Compactification One interesting new feature of timelike compactification is that there is a distinguished point in the moduli space of CFT’s: Proposition 1: ∃! Γ∗ ∈ N for which ¯ open Hclosed = Hopen ⊗ H

(13)

The proof of this is trivial. The string factorizes iff the lattice of zero modes is a direct sum of left- and right-moving lattices. This implies that the left- and †

In general these Lie algebras are not hyperbolic Kac-Moody algebras.

right-moving sublattices are even unimodular. Since the signature of these lattices 1,25 ]. is (1, 25) they are unique. Therefore HΓ∗ = C ⊗ C¯ where C = S(h− 1,25 ) ⊗ C[II 1 Several remarks are in order. The symmetry H (C) is in a sense maximal since the only representation which appears in the physical spectrum is the adjoint representation H 0,1 ⊗ H 1,0 for zero modes of “gauge bosons.” These are states whose existence is dictated by the existence of symmetry. Moreover the form of the string densities is completely fixed by the unbroken symmetry (although the integrals over moduli space typically diverge) [7]. This partially realizes the idea that unbroken fundamental symmetries of strings completely fix the string S -matrix. The unbroken symmetry algebra H 1 (C) is related to the Monster group and has been studied in this context in [1]. Given the remarks in the previous section one might wonder if this “Monster orbit” is dense; one can show that this is not the case. 3.3. Universal Symmetry Let us define a “universal symmetry for toroidal compactifications” to be a Lie algebra L that contains H 1 (HΓ ) for all ESP’s Γ. There is not a unique choice for such a symmetry, but we would like to find one which is natural and, in some sense, minimal. Given the distinguished symmetry of the previous subsection one might expect that H 1 (HΓ∗ ) is a universal symmetry. This is not true. However, there is a very simple construction which does give such a symmetry. First we must put left-movers and right-movers together into a single multiplet: A ρA n = βn ρA = β¯A−26 n

A = 1, . . . 26 A = 27, . . . 52

−n

B [ρA n , ρm ]

=D

AB

nδn+m,0

(14)

Let h26,26 be the corresponding Heisenberg algebra. The commutator equation is clearly Lorentz invariant. We now introduce Γ26,26 , the even unimodular lattice in ˜ 26,26 in 52 dimensions, of signature (26, 26). It is necessary to consider a larger lattice Γ C 52 obtained by allowing integer combinations of purely real or imaginary vectors. Finally we have Proposition 2: Let ˜ = S(h− ) ⊗ C[Γ ˜ 26,26 ] H 26,26

(15)

˜ V ir+ /(H[1] ˜ V ir+ ∩ V ir− · H) ˜ LU ≡ H[1]

(16)

Then the Lie algebra:

˜ V ir denotes restriction to the Virasoro dimension is a universal symmetry. Here H[1] = 1 primaries. +

This construction is closely related to some work of Giveon and Porrati [3] 4. Symmetry-Breaking Physically, time is not compact. Nevertheless, we may try to describe backgrounds with noncompact time as broken symmetry phases of the hyperbolic symmetries of compactified time. The paradigm for this description is the compactification of a single Euclidean coordinate on a circle of radius R. At the self-dual radius (1) (1) R = 1 there is an enhanced affine su(2)L × su(2)R symmetry in the conformal field theory. This leads to a spacetime su(2) × su(2) gauge symmetry. If we increase R to R > 1 then, from the point of view of the spacetime theory, the gauge symmetry is spontaneously broken by the vev of the scalar field ΦR (x) whose zero-mode corre¯ . One way to try to deduce sponds to the modulus of the circle ΦR (p = 0) ↔ ∂X ∂X quantitative consequences of this point of view is to write down the broken Ward identities for the broken symmetries, analogous to the Slavnov-Taylor identities used in the proofs of renormalizability of spontaneously broken gauge theories. Our goal for the remainder of this note will be to write down a set of broken symmetry Ward identities. The results of this section apply to all toroidal compactifications, and are independent of the signature of the compactified dimensions. The basic strategy is quite simple and best explained in the sigma model picture. If E is an ESP and E ′ = E + ∆E is not then we may still relate correlation functions in one theory to those in the other by conformal perturbation theory. Since the action is linear in E this appears to be completely trivial: we simply write R R ′¯ 1 Q Q h i Vi iE ′ = e− 2π ∂XE ∂X i Vi R 1 ¯ Q . = he− 2π ∂X∆E ∂X i Vi iE

(17)

Of course, eq. (17) is much too naive. Conceptually it does not make sense: In CFT we identify states and operators, but the state-spaces of the two conformal field theories HE , HE ′ are different. For example, they are built from different representations of the Virasoro algebra. At best we can hope that conformal per′ turbation theory will define a linear map T E ,E : HE → HE ′ . The manipulations in eq. (17) are also technically wrong because, if we attempt to expand the interaction we will obtain infinity for the integrals of the singular correlators. 4.2. Lorentz Transport In [7] the conformal perturbation series (for toroidal compactifications) is given a precise meaning and the series is summed up to produce a globally welldefined transport on the moduli space of toroidally compactified theories. To state the result we must introduce some notation. Recall that E ∈ B , and via ψ : B ֒→ IH = (O(η) × O(η))\M we can define a ˇ R) on B . (This may be written explicitly as a matrix M¨ right-action of O(D, obius transformation on E .) If we block-decompose the matrix g ∈ O(D) as: g=

g11 g21

g12 g22

(18)

and define natural coordinates on IH: tr tr −1 ∆E(g) = −g21 (g22 ) η = −η(g11 )−1 g12

(19)

we can then state: Proposition 3: Let E ′ = E · g . The integrals in conformal perturbation theory can be defined so that:

− 1 R ∂Y ·∆E·∂Y ¯ Q 2π e

where the transport

−1 R e 2π

¯ ∂Y ·∆E·∂Y

Vi

E

=

E

Y

T g (Vi ) E ′

T g : HE −→ HE ′

(20)

(21)

is defined by T g : |pL ; pR i → |g · (pL ; pR )i ′

A A ρA n → ρn (DgD)A′

(22)

Remark: The operator T g defines a parallel transport on the “bundle of conformal field theories” H → IL, or H → IH. The relation to the Zamolodchikov metric is provided by the projection Π to the space of exactly marginal operators which may be identified with T IH. One can show that

g a a¯ b ¯¯ b ΠT (β−1 β¯−1 )(1)ΠT g (β−1 β−1 )(0) d(∆Ea¯a )d(∆Eb¯b )

(23)

is the right-invariant Zamolodchikov metric T r[G−1 dEG−1 dE tr ]. These remarks are related to the works [8, 9] and to B. Zwiebach’s contribution to this volume. 4.3. Broken Ward Identities Suppose we are at an enhanced symmetry point E , and suppose J is a BRST invariant current. Then the Ward identities associated with J are: 0=

Y XI V j iE h J(z) · V i zi

i

(24)

j6=i

Now consider another background E ′ = E · g = E + ∆E where the symmetry is broken. ¯ does not commute This will happen if the conformal perturbation O = ∂Y · ∆E · ∂Y with the symmetry, that is, if δJ O(z, z¯) = Proposition 4:

I

z

J(w)dwO(z, z¯) 6= 0

(25)

The broken Ward identities are d2 whT g (δJ O)

T g V i (zi )iE ′ + Q P H g µ g i g j h dw (T J )(w, w)T ¯ V )(z , z ¯ ) µ i i i j6=i (T V )iE ′ |w−zi |=ǫ 0=

R

|w−zi |>ǫ

Q

(26)

The proof of this formula is based on an identity ∂ ∂ ¯ z¯)) (T g J(z, z¯)) − (T g J(z, ∂ z¯ ∂z ¯ . The component J¯ may be constructed J µ = (J, J)

T g (δJ O)(z, z¯) = −

(27)

which defines a current explicitly from J and g [7]. The broken Ward identity can be given the following physical interpretation. The vertex operator T g (δJ O)(z, z¯) is the vertex operator for the emission of a zero-momentum goldstone boson πJ (0) for the broken symmetry J . (πJ gets eaten by the vector bosons.) Define δVi by the operator appearing with a simple pole in the ope of T J µ with Vi . Then proposition 4 implies an equality of analytically continued S -matrix elements A A(πJ (0), Vi ) =

X i

A(V1 , . . . δVi , . . . Vn )

(28)

Which may be regarded as a “low-energy theorem.” Thus the same identity can be simultaneously regarded as a statement of explicit symmetry-breaking on the worldsheet and spontaneous symmetry-breaking in spacetime. 5. Conclusion We conclude with two speculations. First, we believe that the broken Ward identities can be used to explain the high energy symmetries of string theory which were the subject of some speculation 5 years ago [5, 11]. We hope this will result from a combination of the results of sections two and four. Second, we think that it is significant that the construction of a universal symmetry for toroidal compactifications involves the “52-dimensional open string.” Many facts about closed string field theory strongly suggest that it is a broken symmetry phase of .... Perhaps it is a broken symmetry phase of some such “open string.” 6. Acknowledgements I would like to thank the Ecole Normale Superieure for hospitality while this note was written. This work is also supported by DOE grant DE-AC02-76ER03075, DOE grant DE-FG02-92ER25121, and by a Presidential Young Investigator Award. 7. References 1. R. Borcherds, “Generalized Kac-Moody Algebras,” J. Algebra 115(1988)501; “Monstrous moonshine and monstrous lie superalgebras,” Invent. Math. 109(1992)405. 2. E. Getzler, “Batalin-Vilkovisky Algebras and Two-Dimensional Toplogical Field Theories” (hep-th/9212043). 3. A. Giveon and M. Porratti, “Duality invariant string algebra and D = 4 effective actions” Nucl.Phys.B355:422-454,1991; “A completely duality invariant effective action of N = 4 heterotic strings,” Phys.Lett.B246:54-60,1990

4. P. Goddard and D. Olive, “Algebras, lattices and strings,” in Vertex Operators in Mathematics and Physics, Proceedings of a Conference eds. J. Lepowsky, S. Mandelstam, I.M. Singer, Springer. 5. D. Gross, “High energy symmetries of string theory,” Phys. Rev. Lett. 60B(1988)1229. 6. B. Lian and G. Zuckerman, “New perspectives on the brst algebraic structure of string theory,” (hep-th/9211072) Toronto-9211072. 7. G. Moore, “Finite in all directions,” hepth/9305139. 8. K.Ranganathan, “Nearby CFT’s in the operator formalism: The role of a connection,” hep-th/9210090 9. K. Ranganathan, H. Sonoda, and B. Zwiebach, “Connections on the StateSpace over Conformal Field Theories,” hep-th/9304053 10. G. Segal, Lectures at the Isaac Newton Institute, August 1992, and lectures at Yale University, March 1993. 11. E. Witten, “Spacetime and topological orbifolds,” Phys. Rev. Lett. 61B(1988)670. 12. E. Witten and B. Zwiebach, “Algebraic Structures and Differential Geometry in 2D String Theory” (hep-th/9201056), Nucl. Phys. B377 (1992) 55. 13. B. Zwiebach, “Closed string field theory: quantum action and the B-V master equation” (hep-th/9206084), Nucl. Phys. B390 (1993) 33.

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close